3.241 \(\int \frac{(1+2 x)^2 (1+3 x+4 x^2)}{\sqrt{2-x+3 x^2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{1}{6} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{11}{27} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{143}{324} (3-2 x) \sqrt{3 x^2-x+2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}} \]

[Out]

(-143*(3 - 2*x)*Sqrt[2 - x + 3*x^2])/324 + (11*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2])/27 + ((1 + 2*x)^3*Sqrt[2 - x +
3*x^2])/6 + (4147*ArcSinh[(1 - 6*x)/Sqrt[23]])/(648*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0989095, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {1653, 832, 779, 619, 215} \[ \frac{1}{6} \sqrt{3 x^2-x+2} (2 x+1)^3+\frac{11}{27} \sqrt{3 x^2-x+2} (2 x+1)^2-\frac{143}{324} (3-2 x) \sqrt{3 x^2-x+2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]

[Out]

(-143*(3 - 2*x)*Sqrt[2 - x + 3*x^2])/324 + (11*(1 + 2*x)^2*Sqrt[2 - x + 3*x^2])/27 + ((1 + 2*x)^3*Sqrt[2 - x +
3*x^2])/6 + (4147*ArcSinh[(1 - 6*x)/Sqrt[23]])/(648*Sqrt[3])

Rule 1653

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*e^(q - 1)*(
m + q + 2*p + 1)), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{(1+2 x)^2 \left (1+3 x+4 x^2\right )}{\sqrt{2-x+3 x^2}} \, dx &=\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{1}{48} \int \frac{(1+2 x)^2 (-44+176 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{1}{432} \int \frac{(1+2 x) (-1716+1144 x)}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}-\frac{4147}{648} \int \frac{1}{\sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}-\frac{4147 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+6 x\right )}{648 \sqrt{69}}\\ &=-\frac{143}{324} (3-2 x) \sqrt{2-x+3 x^2}+\frac{11}{27} (1+2 x)^2 \sqrt{2-x+3 x^2}+\frac{1}{6} (1+2 x)^3 \sqrt{2-x+3 x^2}+\frac{4147 \sinh ^{-1}\left (\frac{1-6 x}{\sqrt{23}}\right )}{648 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0327693, size = 55, normalized size = 0.58 \[ \frac{6 \sqrt{3 x^2-x+2} \left (432 x^3+1176 x^2+1138 x-243\right )-4147 \sqrt{3} \sinh ^{-1}\left (\frac{6 x-1}{\sqrt{23}}\right )}{1944} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + 2*x)^2*(1 + 3*x + 4*x^2))/Sqrt[2 - x + 3*x^2],x]

[Out]

(6*Sqrt[2 - x + 3*x^2]*(-243 + 1138*x + 1176*x^2 + 432*x^3) - 4147*Sqrt[3]*ArcSinh[(-1 + 6*x)/Sqrt[23]])/1944

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 79, normalized size = 0.8 \begin{align*}{\frac{4\,{x}^{3}}{3}\sqrt{3\,{x}^{2}-x+2}}+{\frac{98\,{x}^{2}}{27}\sqrt{3\,{x}^{2}-x+2}}+{\frac{569\,x}{162}\sqrt{3\,{x}^{2}-x+2}}-{\frac{3}{4}\sqrt{3\,{x}^{2}-x+2}}-{\frac{4147\,\sqrt{3}}{1944}{\it Arcsinh} \left ({\frac{6\,\sqrt{23}}{23} \left ( x-{\frac{1}{6}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2-x+2)^(1/2),x)

[Out]

4/3*x^3*(3*x^2-x+2)^(1/2)+98/27*x^2*(3*x^2-x+2)^(1/2)+569/162*x*(3*x^2-x+2)^(1/2)-3/4*(3*x^2-x+2)^(1/2)-4147/1
944*3^(1/2)*arcsinh(6/23*23^(1/2)*(x-1/6))

________________________________________________________________________________________

Maxima [A]  time = 1.54187, size = 108, normalized size = 1.14 \begin{align*} \frac{4}{3} \, \sqrt{3 \, x^{2} - x + 2} x^{3} + \frac{98}{27} \, \sqrt{3 \, x^{2} - x + 2} x^{2} + \frac{569}{162} \, \sqrt{3 \, x^{2} - x + 2} x - \frac{4147}{1944} \, \sqrt{3} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (6 \, x - 1\right )}\right ) - \frac{3}{4} \, \sqrt{3 \, x^{2} - x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2-x+2)^(1/2),x, algorithm="maxima")

[Out]

4/3*sqrt(3*x^2 - x + 2)*x^3 + 98/27*sqrt(3*x^2 - x + 2)*x^2 + 569/162*sqrt(3*x^2 - x + 2)*x - 4147/1944*sqrt(3
)*arcsinh(1/23*sqrt(23)*(6*x - 1)) - 3/4*sqrt(3*x^2 - x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.70268, size = 201, normalized size = 2.12 \begin{align*} \frac{1}{324} \,{\left (432 \, x^{3} + 1176 \, x^{2} + 1138 \, x - 243\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{4147}{3888} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} - x + 2}{\left (6 \, x - 1\right )} - 72 \, x^{2} + 24 \, x - 25\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2-x+2)^(1/2),x, algorithm="fricas")

[Out]

1/324*(432*x^3 + 1176*x^2 + 1138*x - 243)*sqrt(3*x^2 - x + 2) + 4147/3888*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 - x
+ 2)*(6*x - 1) - 72*x^2 + 24*x - 25)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (2 x + 1\right )^{2} \left (4 x^{2} + 3 x + 1\right )}{\sqrt{3 x^{2} - x + 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)**2*(4*x**2+3*x+1)/(3*x**2-x+2)**(1/2),x)

[Out]

Integral((2*x + 1)**2*(4*x**2 + 3*x + 1)/sqrt(3*x**2 - x + 2), x)

________________________________________________________________________________________

Giac [A]  time = 1.18767, size = 85, normalized size = 0.89 \begin{align*} \frac{1}{324} \,{\left (2 \,{\left (12 \,{\left (18 \, x + 49\right )} x + 569\right )} x - 243\right )} \sqrt{3 \, x^{2} - x + 2} + \frac{4147}{1944} \, \sqrt{3} \log \left (-2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} - x + 2}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+2*x)^2*(4*x^2+3*x+1)/(3*x^2-x+2)^(1/2),x, algorithm="giac")

[Out]

1/324*(2*(12*(18*x + 49)*x + 569)*x - 243)*sqrt(3*x^2 - x + 2) + 4147/1944*sqrt(3)*log(-2*sqrt(3)*(sqrt(3)*x -
sqrt(3*x^2 - x + 2)) + 1)